ETL 1110-2-547
30 Sep 95
uncertainty in the values of various parameters used in
A definition of the limit state (e.g., ln (FS) =
calculating some measure of performance, such as the
0).
factor of safety.
A method to estimate the expected value and
(3) In the third context, reliability over a future
standard deviation of the limit state given the
time interval is calculated using parameters developed
expected values and standard deviations of the
from actual data on the lifetimes or frequencies of
parameters (e.g., the Taylor's series or point
failure of similar parts or systems. These are usually
estimate methods).
taken to follow the exponential or Weibull probability
distributions. This methodology is well established in
d. Accuracy of reliability index.
electrical, mechanical, and aerospace engineering
where parts and components routinely require periodic
(1) For rehabilitation studies of geotechnical
replacement. This approach produces a hazard
structures, the reliability index is used as a "relative
measure of reliability or confidence in the ability of a
time period. These functions are used in economic
structure to perform its function in a satisfactory
analysis of proposed geotechnical improvements. The
manner."
development of hazard functions is not part of this
ETL.
(2) The analysis methods used to calculate the
reliability index should be sufficiently accurate to rank
(4) For reliability evaluation of most geotechnical
the relative reliability of various structures and
structures, in particular existing levees, the capacity-
components. However, reliability index values are not
demand model will be utilized, as the question of
absolute measures of probability. Structures,
interest is the probability of failure related to a load
components, and performance modes with higher
event rather than the probability of failure within a
indices are considered more reliable than those with
time interval.
lower indices. Experience analyzing geotechnical
structures will refine these techniques.
c. Reliability index. The reliability index $ is a
measure of the reliability of an engineering system that
reflects both the mechanics of the problem and the
B-3. The Capacity-Demand Model
uncertainty in the input variables. This index was
developed by the structural engineering profession to
a. In the capacity-demand model, the probability
provide a measure of comparative reliability without
of failure or unsatisfactory performance is defined as
having to assume or determine the shape of the
the probability that the demand on a system or
probability distribution necessary to calculate an exact
component exceeds the capacity of the system or
value of the probability of failure. The reliability
component. The capacity and demand can be
index is defined in terms of the expected value and
combined into a single function (the performance
standard deviation of the performance function, and
function), and the event that the capacity equals the
permits comparison of reliability among different
demand taken as the limit state. The reliability R is
structures or modes of performance without having to
the probability that the limit state will not be achieved
calculate absolute probability values. Calculating the
or crossed.
reliability index requires:
b. The concept of the capacity-demand model is
A deterministic model (e.g., a slope stability
illustrated for slope stability analysis in Figure B-1.
analysis procedure).
Using the expected value and standard deviation of the
random variables c and N in conjunction with the
A performance function (e.g., the factor of
Taylor*s series method or the point estimate method,
safety from UTEXAS2).
the expected value and standard deviation of the
factor of safety can be calculated. If it is assumed that
The expected values and standard deviations
the factor of safety is lognormally distributed, then the
of the parameters taken as random variables
natural log of the factor of safety is normally distri-
(e.g., E[N] and FN).
buted. The performance function is taken as the log of
B-2
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